A penalty method for nonparametric estimation of the intensity function ...

Ann. Inst. Statist. Math. Vol. 41, No. 4, 781 807 (1989)

A PENALTY METHOD FOR NONPARAMETRIC ESTIMATION OF THE INTENSITY FUNCTION OF A COUNTING PROCESS ANESTIS ANTONIADIS* Department o f Mathematics, U.E.R. des Sciences, University o f Saint-Etienne, 23, rue du Dr Paul Michelon, 42100 Saint-Etienne, France (Received January 12, 1987; revised June 2, 1988)

Nonparametric estimators are proposed for the logarithm of the intensity function of some univariate counting processes. An Aalen multiplicative intensity model is specified for our counting process and the estimators are derived by a penalized maximum likelihood method similar to the method introduced by Silverman for probability density estimation. Asymptotic properties of the estimators, such as uniform consistency and normality, are investigated and some illustrative examples from survival theory are analyzed. Abstract.

Penalizedlikelihood,counting processes, multiplicative intensity, censored data, martingales, Sobolev spaces, kernel estimation. Key words and phrases:

1.

Introduction

This paper deals with the problem of estimating nonparametrically the intensity function of multiplicative intensity counting processes, which constitute an important class of point processes. The importance of such counting processes is that they provide an alternative to the proportional hazard regression model of Cox (1972). Estimation procedures for the intensity function of point processes have been analyzed in a number of papers, for instance, those written by Ogata (1978), Kutoyants (1979), Lin'kov (1981), Sagalovsky (1983) and Konecny (1984) in the parametric case, Aalen (1975, 1978), Bartoszynski et al. (1981), Leadbetter and Wold (1983), Ramlau-Hansen (1983) and Karr (1987) in the nonparametric case, to cite only few. For the multiplicative intensity model, Aalen (1978) provided an estimator for the cumulative hazard function *This work was conducted while the author was visiting the Department of Mathematics, University of California at lrvine. 781

782

ANESrI-1S A N T O N I A D I S

rather than the intensity function itself. Under appropriate assumptions, Ramlau-Hansen (1983) used kernel function methods from density estimation to smooth the Aaten estimator and to obtain a kernel estimator of the intensity itself in the nmltiplicative intensity model. Karr (1987) used Grenander's method of sieves and obtained estimators for the intensity function that are strongly consistent in the L ~ norm. McKeague (1986) used the sieve method to obtain consistent estimators for the parameters of a general semimartingale regression model which contains the point process multiplicative model as a special case. A common feature of these papers is the use of techniques developed for probability density estimation. In spite of this c o m m o n feature, the means for obtaining detailed results--consistency, asymptotic distribution and the like -depend very much on the specific situation involved. Our interest in the multiplicative intensity counting process model and the questions addressed here was motivated by the work of Bartoszynski et al. (1981). The basic problem they treat is estimation (from i.i.d, realizations) of the intensity function of a nonhomogeneous Poisson process on ~+. The context is metastasis in the growth of malignant tumors. Their techniques are based on penalized maximum likelihood estimation, but questions of consistency or asymptotic normality for their estimators are not considered. Here, we obtain such results for the intensity estimators using methods along the lines of Silverman (1982) for probability density estimation, though some care is needed since the jump times are not i.i.d, random variables. In order to guarantee that the estimators of the intensity are nonnegative, we parametrize the logarithm of the intensity rather than the intensity itself. A penalized maximum likelihood scheme is used to derive the estimators, with its reasonableness .justified by virtue of uniform consistency in probability and asymptotic normality. Penalized maximum likelihood methods have also been applied by O'Sullivan (1983) and by Zucker (1986) for nonparametrically estimating a function representing time-dependent covariate effects in a model extending Cox's original model. The statistical problem we are dealing with is formulated in Section 2, where we also outline some basic results from the theory of counting processes. A survey of this theory intended for applications similar to ours can be found in Gill (1980) or Jacobsen (1982). In Section 3 we present the maximum penalized likelihood procedure for estimating the log intensity and give some conditions for its existence. The remaining sections are devoted to the asymptotic properties of the estimators and to the description of some practical examples. Finally, we would like to point out that our work has been, to a great extent, influenced by Silverman's work on probability density estimation.

ESTIMATING INTENSITY OF A COUNTING PROCESS

2.

783

Formulation of the Aalen model--notation and conventions

Since we are interested in asymptotic properties, we shall consider a sequence of models indexed by n = 1,2,.... As mentioned in the introduction, for each n we will observe an n-component multivariate process

(2.1)

N (') = (N("), N~ ") .... , N,('>) ,

where each c o m p o n e n t of (N(')(t), t e [0, T]) is a counting process over the time interval [0, T]. The sample paths o f N I~l are step functions, zero at time zero, with j u m p s of size + 1 only, no two components j u m p i n g simultaneously. Thus, multiple events cannot occur. In this model, all properties of stochastic processes are relative to a complete right-continuous filtration ( ~ ( , ) : t ~ [0, T]) of sub-a-algebras on the n-th sample space (Q('), o ~ ("), P(n)); O.5~(") represents all the information available up to time t in the n-th model. We will also assume, as in Aalen (1978), that Ep'°'[Ni("I(T)] < oo for each n and every i ( i = l,...,n), and that N (") has a r a n d o m intensity function A (") = ~,~1'~(',~2~('),..., ) 2,(n)) such that (2.2)

2~n)(t) = Yi(n)(t)Oto(t),

where a0 is an u n k n o w n nonnegative deterministic function, called the intensity function, while Y/') are nonnegative observable processes on [0, T], adapted to the filtration ~tt I") and predictable. It will be sufficient for our purpose to note that the adapted processes y t,I are predictable if their sample paths are left-continuous with right-hand limits. By stating that N (') has intensity process A (') we mean that the processes Mg(n) defined by (2.3)

Mfl>(t) = N?)(t) -fo ~>')(u)du

are square integrable orthogonal martingales, with dual predictable projections

(2.4)

(Mi(')' Mi('))(t) = fo 2~'l(u) du .

We shall require that a0 is PtnLalmost all sample paths of limits. Several examples of the above are given in Aalen (1975, applicability. The basic statistical problem

a continuous s m o o t h function and that Yi Inl are left-continuous with right-hand multiplicative intensity model described 1978), which indicate its broad scope of in the multiplicative intensity model (2.2)

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ANESTIS A N T O N I A D I S

is to estimate the intensity function a0 and to discuss asymptotic properties of the estimators as n tends to infinity. Convergence in probability and convergence in distribution will always be relative to the probability measures P(") parametrized by a0 on ~Q("). Our model is nonparametric, meaning that the unknown function a0 is allowed to range freely in some infinite-dimensional space of functions, which will be specified later. In terms of estimation, we have already mentioned in Section 1 that RamlauHansen has obtained a pointwise consistent kernel estimator for a0. However, such a procedure may exhibit considerable fluctuations in practice. With milder assumptions on a0 than ours, McKeague (1986) provided, using the sieve method, an L2-consistent estimator for the intensity function, but did not discuss the asymptotic distribution of his estimator. Our approach is based on a maximum penalized likelihood method which can be realized as a sort of "dual" of the sieve method (see, e.g., Geman and Hwang (1982)).

3. A penalized maximum likelihood method for estimating the intensity In this section, we begin with some general consideration of the basic principle we have adopted for estimation and give conditions which guarantee the existence and properties of our estimator. The notation introduced in Section 2 will be employed throughout.

3.1 The basic principle We shall consider in the sequel the stochastic processes on (Q("), ~'("), P(n)) defined on [0, T] by:

i=l

and

i=l

For each n, let P0~") be a probability measure on (~2C"),~--(~)) that makes the components Ni(~) of N I~) independent homogeneous Poisson processes, each with parameter 1. Then (see, e.g., Aalen (1975), Rebolledo (1978) and Jacobsen (1982)), the distribution of N 0) controls the a m o u n t by which the data are s m o o t h e d to give the estimate. In our case (3.3)

A*x(O) : - forO(s)dN,(s) +fore °''' L(s)ds + 2 J ( 0 ) .

A basic requirement in order to be able to estimate 0 on the whole of [0, T] in a meaningful way is that the processes Yn be strictly positive on [0, T]. If this is not the case, then given a realization of Y,, there will be an infinite n u m b e r of 0(t)'s that give the same value of 21nJ(t). Sometimes, it is clearly possible to make such an assumption, but at other times this may not be possible. Hence, we will introduce a natural modification of the negative likelihood part of (3.3), similar to the one used by R a m l a u - H a n s e n (1983). More precisely, since one may have Y,(t) = 0 for some t in [0, T], we will define

786

(3.4)

ANEST1S ANTONIADIS

l,*,

=

-

f( o(,) Y.(s))

+ f f eO.~L(s)ds,

where ~ ( s ) = I t/vo0/lwith J~(s)[ Yn(s)]-1 = 0, whenever Yn(s) = 0 (the realizations of the processes in(s) are left-continuous functions which may j u m p any time Y,(s) is "updated"). At this point, we should mention that expression (3.2) and the same expression with the log-likelihood function replaced by that in (3.4) do not have the same maximizer with respect to 0. The estimator proposed here deserves consideration for reasons of asymptotic consistency and it is justified by the fact that, asymptotically, it turns out to be a kernel type estimate that adapts locally to the density of the observed j u m p times. As in Aalen (1975), we will assume in the sequel that the processes {~(s)[ Yn(s)] -1, 0 1, O = H~([0, T]) as sets and they have equivalent norms (H~ is the Sobolev space of order m). (iii) The penalty functional J is defined by J(O) = bbH0112 where H is a projection operator on O with finite-dimensional null space O0 containing the constant functions on [0, T]. (iv) There exist positive constants M1 and M2 such that M, 110112 ~

J(O) + II01l~2,t0,rl~

~ Mall0112 .

(v) The u n k n o w n function 00 is smooth in the sense of Wahba (1977), that is, 00 lies in a Sobolev space on [0, T] of order p, where p >_ m. Recall that, when m is an integer, H2m([0, T]) is the Hilbert space of functions on [0, T], whose derivativesf/i) up to the order (m - 1) exist and are absolutely continuous and such t h a t f Ira) is in L 2. The norm of H2m([0, T])


is [Ifll2m = /1~

i--I

ilfli)ll ~2(t0,T1) X/1/2. When ]

787

m is a positive number, one can define

H2m([0, T]) using interpolation theory (see, e.g., Adams (1975)). The spaces H~° are reproducing kernel Hilbert spaces (R.K.H.S.). The set of Assumptions A is designed to represent prior notions about the behavior of the unknown function 00. A(iii) and A(iv) ensure that the penalty functional J is well behaved, and allow the optimization to be numerically tractable. Very often in practice, the penalty functional J(O) is 2 where D: H2m ~ L2([0, T]) is a differential given by J(O)--f0 r (DO)(s)ds operator of order m with O0 = ker (D) (see, e.g., Cox (1984)). Assumption A(v) formulates the fact that it is probably reasonable to believe that 00 is smoother than just being in H2m([0, T]). In the sequel, following Silverman's terminology, the null family of H will be called the family of "infinitely smooth" log intensity functions. The next assumptions pertain to the stochastic part of our model. The conventions of Section 2 are in force. Unless otherwise stated, all limits are taken as n goes to infinity. ASSUMPTIONS B. (i) j,(s)= Ep'°'(a~(s)) converges uniformly to 1 in [0, T] at a rate

o(n-1). (ii) nEe'o,(~(s)[ Y~(s)]-1) is uniformly bounded below and above away from 0 and ~. (iii) {n~(s)[ Y,(s)]- ~; 0 s) > 0, for all 0 < s < T. 3.3

Existence of the MPL estimator A discussion of the existence of the M P L estimator of the intensity function of a n o n h o m o g e n e o u s Poisson process is given in Section 4 of Bartoszynski et al. (1981), drawing on material from Tapia and T h o m p s o n (1978). Since, in that work, the penalty functional is a n o r m (and not a seminorm), the question of existence of our estimator is a little more delicate. The theorem of this section gives a condition for the existence of M P L estimators of the log intensity function and it is similar to T h e o r e m 4.1 of Silverman (1982). However, the context is different and our m e t h o d of proof cannot be thought of as an extension of the techniques of proof used in the aforementioned paper. THEOREM 3.1. Within the notation of this section, the functional An.~, as defined in (3.5) above, has a unique minimizer in 6) whenever there exists a minimizer Oo of l*v(O) in the space of "infinitely smooth" log intensity functions. PROOF. Note first that l'v(0) is weakly continuous and strictly convex on O. Indeed, since, by A s s u m p t i o n A(ii), O is a reproducing kernel Hilbert space, we know that pointwise evaluation is a continuous operator. To establish the weak continuity of l*r(O) on O, we need only to show that for


789

any sequence {Oh, k >_ 1 } converging to 0 in O, fo r e°~(')a~(s) ds ~_~ for e°(S)J, (s) ds. Since the n o r m of 6) is equivalent to a Sobolev n o r m with m >_ 1, Ok ~ 0 uniformly on [0, T]. Now, I J , ( s ) l - < 1 and the exponential function is continuous; thus, the result clearly holds. A straightforward c o m p u t a t i o n shows that the second Gateaux variation (see Tapia and T h o m p s o n (1978)) of l*r(O) at 0 in the direction 11, is given by

"l~v( O)(rl, q) = fro qZ(s) e°(~)L (s) ds , which is P(')-almost surely positive for any t/ in O such that [[r/[[ > 0. Therefore, l*r is positive definite relative to O and by Proposition 16, Appendix 1 of Tapia and T h o m p s o n (1978), l,*,,ris strictly convex in O. The last part of the p r o o f is similar to that given by Cox and O'Sullivan 0985) for the m a x i m u m penalized likelihood of regression functions. More precisely, let 2 > 0 be given and let 01 in O. If B = {0 e O; A,,~(O) 1 in B s u c h that II0kll - ~ as k - ~. Since A,,~(Ok) is bounded, II//(0k)[I must be bounded, so, if Id denotes the identity operator on O, it must be that I[(Id - lI)(Ok)[[ ~ ~ as k -~ ~. Write 1

1

1

0~ = ~ - (Id - II)(Ok) -- - f Ok + --f ( - lI)(Ok). By convexity of l*r we have 1

,

1 l*

l;* r(O~) ----~ - 1;, r(Ok) + ~ . ,

T( --//Ok)

.

However, l*r being strictly convex and having a minimizer in Oo = ker (//), this minimizer is unique and l*r is coercive on O0, i.e., l * r ( O ) ~ oo as II01t-" ~ . Hence, l*v(O~)--" ~ as k - - - ~ . But this implies, by the weak continuity of l*r and the b oundedness of the sequence {[[//(0k)[[ }k>l that l*v(Ok) -" oo as k goes to infinity, which contradicts the definition of (0k)k~_1. []

790

ANESTIS A N T O N I A D I S

4. Asymptotic properties In this section, the consistency and other properties of the MPL estimator introduced above will be studied. Assumptions A and B are in force throughout. 4.1

Some useful facts Note first that, by Assumptions A(ii) and A(v), the intensity function ao(s) = exp (Oo(s)) and its first derivative are functions bounded below and above away from 0 and infinity on [0, T]. Following Utreras (1979a), we consider the following continuous bilinear forms on O × O: B(u, v) = (Hu, o) = (Hu, Ho) and

(4.1)

A (u, v) = for u(t) o(t) ao(t) d t .

The bilinear form A(u,v) is the scalar product of the weighted space L2([0, T], ao(t)dt) and since a0 is bounded away from zero, L2([0, T], dr) and L2([0, T], ao(t)dt) have equivalent norms. Let (/li)i~0 be the eigenvalues of the following variational problem: (4.2)

B(u, v) = 12A (u, v),

A (u, u) = 1 ,

for all o in O with corresponding eigenfunctions 4)o,~bl..... Since B(u, o) = (IIu, Hv),/~ = 0 is an eigenvalue of (4.2) with corresponding eigenspace the finite-dimensional kernel of the projector//. Therefore, the eigenvalue 0 has multiplicity m0 = dim [ker (/7)] (i.e.,/z0 =/Zl . . . . . [Am0-1) and a basis {4)o = 4)~ . . . . . 4'mo-1}of orthonormal eigenfunctions will be chosen in Oo with 4,0 constant on [0, T]. On the orthogonal complement of O0 in O, B and A are symmetric and positive definite, so the eigenvalues (/Zi)i~mo are strictly positive and their corresponding eigenfunctions (~i)i>_dim(Oo)are orthogonal. Moreover, since a0 belongs to H~°([0, T]) and since Assumption A(iv) holds, the eigenvalues associated to (4.2) satisfy the following inequalities: (4.3)

a'iZm 0, let B~ be the symmetric positive definite bilinear form on 6)× 6) defined by


791

B~(u, o) = 2B(u, o) + A(u, o).

(4.4)

It is easy to see that B~ is coercive and defines a n o r m on O which is equivalent to the norm I1" II by A(iv). We now state and prove, when necessary, some technical results which will be used later. The notation f ( h ) ~ g(h) as h ~ 0 means that, for some constants K and K' and some neighborhood V of zero, K 1, Ee,o,{[/~+(T)]2} = Var [/~k(T)] + E2e,",{[fl~(T)]} and Var [/~k(T)] = Ep,",{[/~k(T) - Ee,°,(fl~(T))] 2}

0, B). defined by (4.4) defines on O a norm equivalent to its Sobolev norm. By the Sobolev imbedding theorem (see, e.g., Adams (1975)), for every e > 0, there exists a positive constant C, such that I10- 0o112 I ( 1 + 2/*k)2 -

2C~ [[ k~>lk 1+~

"

/~(T)

22

(1 + 2/./k) 2 q'-

k~+,/t~A(00, 4~k)2 ] Z k>-m°

(1 q- 2]./k) 2

Taking 0 < e < min (2m6, 8m - 3, 2p) one has

E[ Y~ k 1+~ fi2(T) k_>l

]

12-U2m2-1/m

(1 +2/tk) 2 ~ n -

and 22

y~ kl+~ A(Oo, cbk):,u2 = 220(2_Ll+~.+4m_Zpl/2m)= o(2_[l+e_2p]/2m )

k~J

(1 + 2/lk) 2

by Lemmas 4.1 and 4.2. Thus, E [ I I O - 0o11~]

0,

Suppose that

f~) H£(s)I{IH2(s)l 2 ~

0 > e}A~(s)ds ~e`°'

~

as n ~ ~ (we then say that M , satisfies the strong asymptotic rarefaction o f j u m p s property o f the second kind ( S A R J 2 ) ) and that there exists a continuous function F on [0, T], nondecreasing, with F(O) = O, such that

(~¢n, &Cn)(t) =f0' H~(s)A,(s)ds p'"' F ( t ) , as n ~ ~ and all t in [0, T]. Then, there exists a Gaussian process M with m e a n 0 a n d covariance f u n c t i o n F ( m i n (s, t)) on [0, T] 2, such that Mn ~ m in O ([0, T]) as n ~ ~ .

We now reintroduce the sequences 0 and 0 of the previous section and we assume that, as n ~ ~ , 2 ~ 0 and obeys the conditions stated in Theorems and Lemmas of Section 4. Under such conditions, 0 - 0 converges uniformly to 0 in probability as n ~ ~ . Therefore, for any s in [0, T], 0(s) and O(s) will have the same asymptotic distribution, if such a distribution exists. To derive the result we will use the fact that 0 behaves as a kernel smoothing estimator with a particular kernel. More precisely, let, for each t in [0, T], Qt be the functional defined on O by Q,(O)

=

~ foT[o(m)(s)]2ds + ~ foT02(S)OlO(S)ds -

O(t) ,

where from now on we assume that [IHO II ~ ~ = L r (o(rnJ(s))2ds.Since a0 and its first (m - 1) derivatives are bounded on [0, T], the functionals Qt have a unique minimizer in O, say K~(., t). Moreover, for each 2 > 0, Q~(O) + O(t) defines a norm on O equivalent to its Sobolev norm. Therefore, as in Cox (1984), K~(., .) is the Green's function of a linear elliptic boundary value


799

problem and it is symmetric. In fact, {Ka(s, t); (s, t) e [0, T] 2} is the reproducing kernel of the R.K.H.S. O~, where O~ denotes the space O with the n o r m Qt(O) + O(t). It is straightforward to see that our approximating estimator for the log intensity function 00 is implicitly given, for all s in [0, T] by: (5.1)

~(s) = forK~(s, t)Oo(t)ao(t)dt - forK~(s, t)[ L(t) - 1]ao(t)dt r L(t) + fo K a ( s ' t ) ~ ( t ) dM"(t)"

Our consistency result will follow from the particular structure of the kernel Ka. Let x* be the symmetric kernel function on ~ given for each x>0by tc*(x) + I¢(x) = 2 '/2 exp ( - x~ V/2) cos ( x / x / ~ ) , where ~c denotes the solution vanishing at + ~ and - ~ of

(5.2)

( -- 1)mK(2ml + K = d ,

with ~ denoting the Dirac delta function. Appealing to Theorem B, Proposition 2 and Remarks in Section 5 of Silverman (1984), there exists a kernel function Kc on ~, depending on both x* and x, such that, for every t e [0, T] one has

(5.3)

sup 1~l/2mao(t)-l/2raKa(l + ~l/2muo(t)-l/2mx, l) x~ ~,(~o)

Kc(X) I do(t)

0 sufficiently small, we have, as

IlO- 011~=

Op[2-c/2m( n 12-1/m ~- 2(2p-lj/2m)] .

Hence, for all t ~ [0, T] and as n -- oo, V/n21/4m(O(t)

-- O(l)) = Oe[2-~/2m(n - 1/22-3/4m -~- 2 (4p- ll/4rn)] ,

Since r/I2/3)m-~2 --~ oc as n ~ oo, for e sufficiently s m a l l n~/221/4m(o(1) - O(l)) converges to 0 in probability as n ~ ~ . Therefore, n~/221/4"(O(t) - Oo(t)) and n~/22~/4'~(O(t) - Oo(t)) will have the same asymptotic distribution. By L e m m a 4.2 and the rate at w h i c h 2--" 0, it is c l e a r t h a t n~/221/4m(E[O(t)]Oo(t))~ 0 as n ~ oo. Thus, nl/221/nm(o(t)- E[0(t)]) will have the same asymptotic distribution as n~/22'/4m(O(t)- Oo(t)). Using expression (5.3), the random part of n~/22~/4"(O(t) - E[0(t)]) is given by (5.4)

T 1/4m L(t) f~ x/n2 K~(s,t)~(t)dM~(t),

s~ [0, T ] .

Set Hn(t) = W~21/4mg2(s, t)(~(t)/Yn(t)) with s fixed in [0, T]. Since K~(s, t) is continuous in t, {H,} defines a sequence of predictable processes. In order to prove our result, it remains to check the conditions of Proposition 5.1 above. We have, for every e > 0,

Z(t) {

= n2

l/4m

--=--,x/nn~(t)

IK~(s,t)l r n ( t ) > e v

}

Let Vn be defined by

Vn(t) =/],1/4,,. x

(S-

t)ao(t) I/2m ] 2_1/2m L(t) 21/2m • • ao(t) l/2m-l" n ~',(t)

"

By the uniform approximation (5.3) and its rate with respect to 2, I{t Hn(t)[ > e} and I{] Vn(t)[ > ~n 1/2} have the same asymptotic behavior. Now, xc is uniformly bounded in ~ and a0 is logarithmically uniformly bounded on [0, T]. By Assumption B(iii), nJn]/l converges uniformly in probability towards a continuous function bounded on [0, T]. Hence, I{1V.(t)[ > en 1/2} ~ 0 uniformly on [0, T], since H1/22 l/4m ~ oo as n -- ~. By definition of H~,

ESTIMATING INTENSITY OF A COUNTING PROCESS T

801

2

fo H,; (s)I{ IHn (s) l > e} Y. (s) ao (s) ds - 7 0 , as n --- ~ and the S A R J 2 is satisfied. Again, by (5.3), one has: (5.5)

r~-2,_ L(I) forH~(t)An(t)dt =J0( T eta11/2m /~a ~a, t) - ~ ao(t)dt

= f T t ~ 1/2m K2[( S -- t) ;- l/2m (~0(1)1/2m] •" 0 t]l/mao(i )- 1/m(l,o(t)2

o(0

L(t)

dt

" L(t) + Oe(al/")f ° n ~ ( t ) ao(t)dt.

The last term on the right-hand side of (5.5) converges to 0 in probability, since 2 ~ 0 as n - ~ ~ ; therefore expression (5.5) has the same limit in probability as T~ - 1/2m K2[( S -- t)~-l/2m ao(1)l/2m]

fo

CtO(1)- I/m flo( t) 2

L(t)

n ~ ( t ) dt.

This last expression, by a change of variables, is equal to

f(~-r);..... n L ( s (5.6)

as 2 ....

/,//]l/2m)

K2 [uao(s- u; 1/2m) 1/2rn]

~n(S : ~

O[O('----S-~ Ig"-~,/2re)l-1/--""~ rill.

Since nJn Yn' converges uniformly in probability on [0, T] and the limit is a continuous function, and since xc2 and ao l÷l/m are uniformly continuous and b o u n d e d on [0, T], expression (5.6) converges, as n -, ~ , to

ao(s)

Xc[Uao(s)

]du-

ao(S),_,/2m _

.

Now, Theorem 5.1 follows from Proposition 5.1. []

Remarks. 1. By using a multivariate extension of Rebolledo's theorem, given in Appendix I of the paper by Andersen and Gill (1982), and the CramerWold device, it is not difficult to see that the finite-dimensional distributions of {0(s), s e [0, T]} are asymptotically multivariate Gaussian. 2. Some rates of uniform consistency have been obtained by RamlauHansen (1983) for the kernel estimator of the u n k n o w n intensity; though his results are for different estimators than ours, they appear to be weaker insofar as a comparison is possible (see Ramlau-Hansen (1983), Theorem

802

ANESTIS ANTONIADIS

4.1.3 and the rates given in the proof of Theorem 5.1 above).

6. Examples This section describes some typical statistical models for which the penalized likelihood procedure of the previous sections can be applied.

Example 6.1. (Hazard rate for a random censored i.i.d, sample) In statistical analysis of survival data (or failure time data) from a homogeneous population, one is interested in estimating the death intensity (or force of mortality, or hazard function). Nonparametric estimation about hazard rates has been studied by several authors. For recent references, see for example Jacobsen (1982), Yandell (1983), Tanner and Wong (1983) and Ramlau-Hansen (1983), where kernel estimators are defined, extending the results of Rice and Rosenblatt (1976) for kernel estimators in the absence of censoring. Consider the analysis of a sample of n individuals. The individuals under study may consist of patients at a given hospital suffering from some lethal disease. Let XI, X2,..., Xn denote nonnegative independent, identically distributed random lifetimes (times to failure) for the n individuals under study, and independent of the Xi's, let C1, C2,..., Cn be the corresponding censoring times. In this kind of data collection, right censoring is inevitable, since in practice one cannot continue the data collection until all individuals are observed to die. Censoring may also occur because some individuals are lost from followup. Suppose that the distribution function F of the Xi's is absolutely continuous with corresponding density f and that H, the continuous distribution function of the C?s, are such that F ( T ) < 1 and H ( T - ) < 1. The observed random variables consist of i.i.d, triples (Z~, ~;, Y,.) where Zi = min (Xg, Ci) is the observable portion of the i-th individual's lifetime, Oi = l~x,~_c,) and Y; is defined by Y-(t)= l~z,>,). Now let Ni(t) denote the indicator function of an uncensored failure for individual i prior to time t: Ni(t) = I(z, 2). Hence, no two component processes will jump, with probability one, at the same time. The number of failures at time t or earlier is given by


803

and has a stochastic intensity process

Am(t) = ? . ( t ) a o ( O , where n

~(t)

= i~=ll{Xi>_t, Ci>_,) =

I n -- ) g n ( / - ) ] +

denotes the n u m b e r of individuals alive just before time t (risk set size at time t-). Note that Yn(t) is binomially distributed with parameters n and [(1 - F(t))(l - H(t-))] (Aalen (1975), L e m m a 4.2). Thus, j , ( t ) = E(J~(t)) = 1 - [(1 - F(t))(1 - H(t-))]" andj~ converges uniformly to 1 on [0, T] at an exponential rate as n - - - ~ . Thus, Assumption B(i) of Section 3 holds. Moreover, since 0